
where \(g_*\)  and \(g_s\)  denote the effective degrees of freedom related to energy and entropy density, respectively.
In principle, the functional forms
\(g_*, g_s\)  could have an extra
\(k\) -dependence through the respective \(\theta _k\)  parameters, but the
overall impact is expected to be minor and will be disregarded in the following.
Moreover, we assume here that thermally produced axions give
a negligible contribution to \(N_{\rm eff}\) . This is typically
the case for \(f_a \gtrsim 10^7\)  GeV [1]}, [2]}, [3]}.
